Anderson localization: from single particle to many body problems.

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## Anderson localization: from single particle to many body problems.

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**Anderson localization:from single particle to many body**problems. (4 lectures) Igor Aleiner ( Columbia University in the City of New York, USA ) Windsor Summer School, 14-26 August 2012**Lecture # 1-2 Single particle localization**Lecture # 2-3 Many-body localization**extended**localized Summary of Lectures # 1,2 • Conductivity is finite only due to broken translational invariance (disorder) • Spectrum (averaged) in disordered system is gapless (Lifshitz tail) • Metal-Insulator transition (Anderson) is encoded into properties of the wave-functions Metal Insulator**Distribution function of the local densities of states is**the order parameter for Anderson transition • Interference corrections due to closed loops are singular; For d=1,2 they diverges making the metalic phase of non-interacting particles unstable; • Finite T alone does not lift localization; metal insulator**Interactions at finite T lead to finite**• System at finite temperature is described as a good metal, if , in other words • For , the properties are well described by ??????**Lecture # 3**• Inelastic transport in deep insulating regime • Statement of many-body localization and many-body metal insulator transition • Definition of the many-body localized state and the many-body mobility threshold • Many-body localization for fermions (stability of many-body insulator and metal)**Inelastic processes: transitions between localized states** energy mismatch (inelastic lifetime)–1 (any mechanism)**Without Coulomb gap**A.L.Efros, B.I.Shklovskii (1975) Phonon-induced hopping energy difference can be matched by a phonon Variable Range Hopping Sir N.F. Mott (1968) Mechanism-dependent prefactor Optimized phase volume Any bath with a continuous spectrum of delocalized excitations down tow = 0 will give the same exponential**⟶ 0 ????? (All one-particle states are localized)**Drude Electron phonon Interaction does not enter “metal” “insulator”**Q:Can we replace phonons with e-h pairs and obtain**phonon-lessVRH? Drude Electron phonon Interaction does not enter “metal” “insulator”**Q:Can we replace phonons with e-h pairs and obtain**phonon-lessVRH? A#1: Sure Easy steps: Person from the street (2005) (2005-2011) 1) Recall phonon-less AC conductivity: Sir N.F. Mott (1970) 2) Calculate the Nyquist noise (fluctuation dissipation Theorem). 3) Use the electric noise instead of phonons. 4) Do self-consistency (whatever it means).**Q:Can we replace phonons with e-h pairs and obtain**phonon-lessVRH? A#1: Sure [Person from the street (2005)] A#2: No way [L. Fleishman. P.W. Anderson (1980)] (for Coulomb interaction in 3D – may be) Thus, the matrix element vanishes !!! is contributed by rare resonances R g d 0 * **Metal-Insulator Transition and many-body**Localization: [Basko, Aleiner, Altshuler (2005)] and all one particle state are localized Drude metal insulator (Perfect Ins) Interaction strength**Many-body mobility threshold**[Basko, Aleiner, Altshuler (2005)] metal insulator All STATES EXTENDED • many-body • mobility threshold All STATES LOCALIZED Many body DoS**“All states are localized“**means Probability to find an extended state: System volume**Many body localization means any excitation is localized:**Extended Localized**States always thermalized!!!**All STATES EXTENDED All STATES LOCALIZED Entropy States never thermalized!!! Many body DoS**Is it similar to Anderson transition?**Why no activation? Many body DoS One-body DoS**Physics: Many-body excitations turn out to be localized in**the Fock space**Fock space localization in quantum dots (AGKL, 1997)**´ e e e e ´ ´ ´ e ´ No spatial structure ( “0-dimensional” ) ´ - one-particle level spacing;**Fock space localization in quantum dots (AGKL, 1997)**5-particle excitation 1-particle excitation 3-particle excitation Cayley tree mapping**Fock space localization in quantum dots (AGKL, 1997)**5-particle excitation 1-particle excitation 3-particle excitation • Coupling between states: • Maximal energy mismatch: • Connectivity: - one-particle level spacing;**Metal-Insulator “Transition” in zero dimensions**[Altshuler, Gefen, Kamenev,Levitov (1997)] In the paper: Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] metal insulator Interaction strength**Metal-Insulator “Transition” in zero dimensions**[Altshuler, Gefen, Kamenev,Levitov (1997)] - one-particle level spacing; Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] 1-particle level spacing in localization volume; • Localization in Fock space • = Localization in the coordinate space. • 2) Interaction is local;**Metal-Insulator “Transition” in zero dimensions**[Altshuler, Gefen, Kamenev,Levitov (1997)] - one-particle level spacing; Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] 1-particle level spacing in localization volume; 1,2) Locality: 3) Interaction matrix elements strongly depend on the energy transfer, w:**Matrix elements:**???? In the metallic regime:**Matrix elements:**???? In the metallic regime: 2**Matrix elements:**???? 2**Metal-Insulator “Transition” in zero dimensions**[Altshuler, Gefen, Kamenev,Levitov (1997)] - one-particle level spacing; Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] 1-particle level spacing in localization volume; 1,2) Locality: 3) Interaction matrix elements strongly depend on the energy transfer, w:**Effective Hamiltonian for MIT.**We would like to describe the low-temperature regime only. Spatial scales of interest >> 1-particle localization length Otherwise, conventional perturbation theory for disordered metals works. Altshuler, Aronov, Lee (1979); Finkelshtein (1983) – T-dependent SC potential Altshuler, Aronov, Khmelnitskii (1982) – inelastic processes**Reproduces correct behavior of the**tails of one particle wavefunctions No spins**j1**l1 l2 j2 Interaction only within the same cell;**random**signs Parameters:**What to calculate?**Idea for one particle localizationAnderson, (1958); MIT for Cayley tree: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) – random quantity No interaction: Insulator Metal**Probability Distribution**Note: metal insulator Look for:**Iterations:**Cayley tree structure**Nonlinear integral equation with random coefficients**Decay due to tunneling after standard simple tricks: Decay due to e-h pair creation + kinetic equation for occupation function**Stability of metallic phase**Assume is Gaussian: 2 ( ) >>**Probability Distributions**“Non-ergodic” metal**Kinetic Coefficients in Metallic PhaseWiedemann-Frantz law ?****Trouble !!!**So far, we have learned: Non-ergodic+Drude metal Insulator ???**Stability of the insulator**Nonlinear integral equation with random coefficients is a solution Notice: for Linearization:**metal**# of interactions # of hops in space Recall: insulator STABLE h probability distribution for a fixed energy unstable**So, we have just learned:**Non-ergodic+Drude metal Metal Insulator**Estimate for the transition temperature for general case**1) Start with T=0; 2) Identify elementary (one particle) excitations and prove that they are localized. 3) Consider a one particle excitation at finite T and the possible paths of its decays: Interaction matrix element # of possible decay processes of an excitations allowed by interaction Hamiltonian; Energy mismatch**Summary of Lecture # 3:**• Existence of the many-body mobility threshold is established. • The many body metal-insulator transition is not a thermodynamic phase transition. • It is associated with the vanishing of the Langevine forces rather the divergences in energy landscape (like in classical glass) • Only phase transition possible in one dimension (for local Hamiltonians) Detailed paper: Basko, I.A.,Altshuler, Annals of Physics 321 (2006) 1126-1205 Shorter version: …., cond-mat/0602510; chapter in “Problems of CMP”